Paper Info
Title
The Convergence of Generalized Lanczos Methods for Large Unsymmetric Eigenproblems
Abstract
In this paper, we investigate the convergence theory of generalized Lanczos methods for solving the eigenproblems of large unsymmetric matrices. Bounds for the distances between normalized eigenvectors and the Krylov subspace ${\cal K}_m(v_1,A)$ spanned by $v_1, Av_1, \ldots, A^{m-1}v_1$ are established, and a priori theoretical error bounds for eigenelements are presented when matrices are defective. Using them we show that the methods will still favor the outer part eigenvalues and the associated eigenvectors of $A$ usually though they may converge quite slowly in the case of $A$ being defective. Meanwhile, we analyze the relationships between the speed of convergence and the spectrum of $A$. However, a detailed analysis exposes that the approximate eigenvectors, Ritz vectors, obtained by generalized Lanczos methods for any unsymmetric matrix cannot be guaranteed to converge in theory even if approximate eigenvalues, Ritz values, do. Therefore, generalized Lanczos algorithms including Arnoldi's algorithm and IOMs with correction are provided with necessary theoretical background.
Year
DOI
Venue
1995
10.1137/S0895479893246753
SIAM J. Matrix Analysis Applications
Keywords
Field
DocType
large unsymmetric eigenproblems,convergence theory,ritz vector,approximate eigenvectors,approximate eigenvalues,normalized eigenvectors,generalized lanczos methods,associated eigenvectors,large unsymmetric matrix,generalized lanczos method,ritz value,generalized lanczos algorithm,orthogonal projection,derivative,orthonormal basis,chebyshev polynomial
Krylov subspace,Lanczos approximation,Mathematical optimization,Lanczos resampling,Matrix (mathematics),Arnoldi iteration,Mathematical analysis,Lanczos algorithm,Orthonormal basis,Mathematics,Eigenvalues and eigenvectors
Journal
Volume
Issue
ISSN
16
3
0895-4798
Citations 
PageRank 
References 
16
2.04
0
Authors
1
Name
Order
Citations
PageRank
Zhongxiao Jia112118.57